Problem 21 - Calculus and its applications

Question

Let \(x\) be a continuous random variable with a standard normal distribution. Using Table A, find each of the following. $$ P(x \geq 3.01) $$

Step-by-Step Solution

Answer

The probability \(P(x \geq 3.01)\) is approximately 0.0013.

Step 1: Understanding the Problem

The problem requires us to find the probability of a standard normal variable \(x\) being greater than or equal to 3.01. Since \(x\) follows a standard normal distribution, it has a mean of 0 and a standard deviation of 1.

Step 2: Consult the Standard Normal Distribution Table

We need to use the standard normal distribution table (Table A) to find \(P(z \geq 3.01)\). These tables typically provide \(P(z \leq a)\), where \(a\) is a specified value, so we will need to adjust our approach accordingly.

Step 3: Calculate \(P(z \leq 3.01)\)

Look up the value 3.01 in the standard normal distribution table. The table gives \(P(z \leq 3.01)\), which represents the probability that a standard normal variable \(z\) is less than or equal to 3.01.

Step 4: Utilize Complement Rule to Find \(P(z \geq 3.01)\)

Use the complement rule: \(P(z \geq 3.01) = 1 - P(z \leq 3.01)\). Subtract the value obtained in the previous step from 1 to find the probability that \(z\) is greater than or equal to 3.01.

Step 5: Retrieve the Table Value and Calculate Final Probability

Assume the standard normal table indicates \(P(z \leq 3.01) = 0.9987\). Thus, \(P(z \geq 3.01) = 1 - 0.9987 = 0.0013\).

Key Concepts

Probability Computation

Understanding how to compute probability in the context of a standard normal distribution is a fundamental statistics skill. In a standard normal distribution, the data is distributed in a bell curve where the mean is zero, and the standard deviation is one. Probability in this context refers to the likelihood that a particular value or a range of values will occur. For continuous random variables, such as those in the standard normal distribution, the probability is calculated over intervals.

When faced with a problem like finding the probability that a random variable, say \(x\), is greater than or equal to a particular value, you need the cumulative probability up to that value and may need to use additional rules or transformations to solve it. These probabilities are often found using standard normal distribution tables, which help us understand where a specific value lies on the normal curve.

• The concept involves determining areas under the curve of the standard distribution.
• Probabilities are directly tied to these areas.
• Knowing exact values of standard normal distribution allows for calculating specific event probabilities.

Complement Rule

In probability theory, the complement rule is a useful tool—especially when calculating standard normal probabilities. It's employed to find the probability of the occurrence of an event by knowing the probability of its complement. In the standard normal distribution, calculating the probability of being greater than or equal to a given value becomes manageable with this rule.

For most standard normal distribution tables, you will find the cumulative probability that a variable is less than or equal to a certain value—i.e., \(P(z \leq a)\). To find the probability that a variable is instead greater than or equal to a value, \(P(z \geq a)\), you make use of the complement rule:

• The complement of \(P(z \geq a)\) is \(P(z < a)\).
• Therefore, \(P(z \geq a) = 1 - P(z < a)\).

In the context of our problem, if \(P(z \leq 3.01) = 0.9987\), then using the complement rule, \(P(z \geq 3.01) = 1 - 0.9987 = 0.0013\). This rule is extremely helpful in cases where the table provides only one side of probability, optimizing your calculations effectively.

Standard Deviation

Standard deviation is a measure that indicates the amount of variability or spread in a set of values. In a standard normal distribution, the standard deviation is always 1, which signifies that data points are typically one unit away from the mean on average.

This provides the basis for transforming any normal random variable into a standard normal variable (z-score), which makes it easier to use probability distribution tables. Understanding the role of standard deviation in a standard normal distribution is vital for interpreting probabilities and predictions effectively.

With a standard deviation of 1, all data manipulations or predictions made within the standard normal distribution become uniform and more straightforward. This simplifies the analysis, allowing for easy comparisons across various datasets or normal variables:

• Standard deviation links the data values to the mean.
• It turns real values into standard scores (z-scores), facilitating easy usage of distribution tables.
• Knowing it's standardized to 1 makes it a universally applicable benchmark for z-score meaning and interpretation.

In practice, standard deviation gives meaningful insight into the role each observation plays within the dataset, further enhancing the grasp of probability computations.